Section 1.1 Sets of Numbers and Interval Notation

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Review of Basic Algebraic Concepts
1.1 Sets of Numbers and Interval Notation 1.2 Operations on Real Numbers 1.3 Simplifying Expressions 1.4 Linear Equations in One Variable 1.5 Applications of Linear Equations in One Variable 1.6 Literal Equations and Applications to Geometry 1.7 Linear Inequalities in One Variable 1.8 Properties of Integer Exponents and Scientific Notation
In Chapter 1, we present operations on real numbers, solving equations, and applications. This puzzle will help you familiarize yourself with some basic terms and geometry formulas. You will use these terms and formulas when working the exercises in Sections 1.5 and 1.6.

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Across
4. Given that distance rate time, what is the distance from Oklahoma City to Boston if it takes 28 hr of driving 60 mph? 5. What is the sum of the measures of the angles of a triangle? 6. What number is twice 6249? 8. If the measure of an angle is 48º, what is the measure of its supplement? 10. If a rectangle has an area of 8670 ft2 and a width of 85 ft, what is its length?

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Down
1. What is the next consecutive 2. If the measure of an angle is complement? 3. What is 25% of 25,644? 5. What is the next consecutive 7. What is 10% of 87,420? 9. What is the next consecutive even integer after 4308? 2º, what is the measure of its
8 9

integer after 1045?
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odd integer after 3225?

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Chapter 1 Review of Basic Algebraic Concepts

Section 1.1
Concepts
1. 2. 3. 4. 5. Set of Real Numbers Inequalities Interval Notation Union and Intersection of Sets Translations Involving Inequalities

Sets of Numbers and Interval Notation
1. Set of Real Numbers
Algebra is a powerful mathematical tool that is used to solve real-world problems in science, business, and many other fields. We begin our study of algebra with a review of basic definitions and notations used to express algebraic relationships. In mathematics, a collection of elements is called a set, and the symbols { } are used to enclose the elements of the set. For example, the set {a, e, i, o, u} represents the vowels in the English alphabet. The set {1, 3, 5, 7} represents the first four positive odd numbers. Another method to express a set is to describe the elements of the set by using set-builder notation. Consider the set {a, e, i, o, u} in set-builder notation. description of set

Set-builder notation: {x | x is a vowel in the English alphabet}

“the set of”

“all x ”

“such that”

“x is a vowel in the English alphabet”

Consider the set {1, 3, 5, 7} in set-builder notation. description of set

Set-builder notation: {x | x is an odd number between 0 and 8}

“the set of ”

“all x ”

“such that”

“x is an odd number between 0 and 8”

Several sets of numbers are used extensively in algebra. The numbers you are familiar with in day-to-day calculations are elements of the set of real numbers. These numbers can be represented graphically on a horizontal number line with a point labeled as 0. Positive real numbers are graphed to the right of 0, and negative real numbers are graphed to the left. Each point on the number line corresponds to exactly one real number, and for this reason, the line is called the real number line (Figure 1-1).
5 4 3 2 1 0 1 2 3 4 Negative numbers Positive numbers Figure 1-1 5

Several sets of numbers are subsets (or part) of the set of real numbers. These are The The The The The set set set set set of of of of of natural numbers whole numbers integers rational numbers irrational numbers

Section 1.1 Sets of Numbers and Interval Notation

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Definition of the Natural Numbers, Whole Numbers, and Integers
The set of natural numbers is {1, 2, 3, . . . }. The set of whole numbers is {0, 1, 2, 3, . . . }. The set of integers is { . . . , 3, 2, 1, 0, 1, 2, 3, . . . }. The set of rational numbers consists of all the numbers that can be defined as a ratio of two integers.

The set of rational numbers is 5 q 0 p and q are integers and q does not equal zero}. p Definition of the Rational Numbers

Example 1

Identifying Rational Numbers

Show that each number is a rational number by finding two integers whose ratio equals the given number. a. 4 7 b. 8 c. 0.6 d. 0.87

Solution:

a.

is a rational number because it can be expressed as the ratio of the integers 4 and 7.

4 7

b. 8 is a rational number because it can be expressed as the ratio of the integers 8 and 1 18 8 2. In this example we see that an integer is also a 1 rational number. c. 0.6 represents the repeating decimal 0.6666666 and can be expressed as the ratio of 2 and 3 10.6 2 2. In this example we see that a repeating 3 decimal is a rational number.
87 d. 0.87 is the ratio of 87 and 100 10.87 100 2. In this example we see that a terminating decimal is a rational number.

TIP: Any rational number can be represented by a terminating decimal or by a repeating decimal.

Skill Practice

Show that the numbers are rational by writing them as a ratio of integers. 2. 0 3. 0.3 4. 0.45

1. –9 8

Some real numbers such as the number p (pi) cannot be represented by the ratio of two integers. In decimal form, an irrational number is a nonterminating, nonrepeating decimal. The value of p, for example, can be approximated as p 3.1415926535897932. However, the decimal digits continue indefinitely with no pattern. Other examples of irrational numbers are the square roots of nonperfect squares, such as 13 and 111.

The set of irrational numbers is {x 0 x is a real number that is not rational}. Note: An irrational number cannot be written as a terminating decimal or as a repeating decimal.

Definition of the Irrational Numbers

Skill Practice Answers
9 8 1 3. 3 1. 0 1 45 4. 100 2.

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Chapter 1 Review of Basic Algebraic Concepts

The set of real numbers consists of both the rational numbers and the irrational numbers. The relationships among the sets of numbers discussed thus far are illustrated in Figure 1-2.
Real numbers Rational numbers
2 7

0.25 0.3

Irrational numbers √2 √17

Integers … 3, 2, 1 Whole numbers 0 Natural numbers 1, 2, 3, …

Figure 1-2

Example 2

Classifying Numbers by Set

Check the set(s) to which each number belongs. The numbers may belong to more than one set.
123 6
2 7

Natural Numbers

Whole Numbers

Integers

Rational Numbers

Irrational Numbers

Real Numbers

3 2.35

Solution:
123 6
2 7

Natural Numbers

Whole Numbers

Integers